A displacement vector is a mathematical object that indicates how far and in what direction a point has moved from an initial position to a final position. In the context of mathematics it is represented as a vector that connects two points; in diagrams it is often drawn as an arrow from the starting point to the ending point.

Definition and basic properties

Formally, the displacement Δr equals the final position vector r2 minus the initial position vector r1: Δr = r2 − r1. This vector encodes both magnitude (the shortest distance between the points) and direction. Unlike the total path length, displacement depends only on the endpoints and not on the route taken. A displacement is a type of vector, so it follows the usual rules for addition and scalar multiplication.

Components and notation

In coordinate form a displacement in the plane can be written as Δr = (Δx, Δy), where Δx = x2 − x1 and Δy = y2 − y1. In three dimensions the vector has a third component Δz. The magnitude |Δr| = sqrt(Δx² + Δy² [+ Δz²]) gives the straight-line distance. Graphically the vector is a directed segment, often described as the shortest or straight line between the two positions.

Uses in physics and kinematics

Displacement is fundamental in physics, particularly in mechanics and kinematics. Average velocity is defined as the displacement divided by the time interval, so that two motions with the same distance but different displacements have different average velocities. Displacement also appears when relating motion to other quantities such as speed (a scalar) and acceleration (a vector derived from velocity changes).

Examples and important distinctions

  • Walking 100 m east then 100 m west yields a total distance of 200 m but a net displacement of zero.
  • Driving around a circuit can produce a large distance traveled while the displacement between start and finish may be small.
  • Displacement vectors add tip-to-tail: two successive displacements Δr1 and Δr2 combine to Δr1 + Δr2.

Understanding displacement clarifies many basic concepts in motion: it separates net change from accumulated path length, provides a direction-aware measure of relocation, and integrates naturally with vector operations used throughout applied mathematics and physics.